$\dfrac{ 9t - 6u }{ -7 } = \dfrac{ -t - 10v }{ 5 }$ Solve for $t$.
Solution: Multiply both sides by the left denominator. $\dfrac{ 9t - 6u }{ -{7} } = \dfrac{ -t - 10v }{ 5 }$ $-{7} \cdot \dfrac{ 9t - 6u }{ -{7} } = -{7} \cdot \dfrac{ -t - 10v }{ 5 }$ $9t - 6u = -{7} \cdot \dfrac { -t - 10v }{ 5 }$ Multiply both sides by the right denominator. $9t - 6u = -7 \cdot \dfrac{ -t - 10v }{ {5} }$ ${5} \cdot \left( 9t - 6u \right) = {5} \cdot -7 \cdot \dfrac{ -t - 10v }{ {5} }$ ${5} \cdot \left( 9t - 6u \right) = -7 \cdot \left( -t - 10v \right)$ Distribute both sides ${5} \cdot \left( 9t - 6u \right) = -{7} \cdot \left( -t - 10v \right)$ ${45}t - {30}u = {7}t + {70}v$ Combine $t$ terms on the left. ${45t} - 30u = {7t} + 70v$ ${38t} - 30u = 70v$ Move the $u$ term to the right. $38t - {30u} = 70v$ $38t = 70v + {30u}$ Isolate $t$ by dividing both sides by its coefficient. ${38}t = 70v + 30u$ $t = \dfrac{ 70v + 30u }{ {38} }$ All of these terms are divisible by $2$ $t = \dfrac{ {35}v + {15}u }{ {19} }$